D and E are the mid-points of the sides AB and AC of DABC and O is any point on side BC. O is joined to A. If P and Q are the mid-points of OB and OC respectively, then DEQP is (A) a square (B) a rectangle (C) a rhombus (D) a parallelogram

#NCERT
#Quadrilaterals
#Exemplar Maths for Class 9
#Maths

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D and E are the mid-points of the sides AB and AC of DABC and O is any point on side BC. O is joined to A. If P and
Q are the mid-points of OB and OC respectively, then DEQP is
(A) a square (B) a rectangle
(C) a rhombus (D) a parallelogram

Answers (1)

By midpoint theorem
$DE\parallel BC$   …..(1)
i.e.,   $DE=\frac{1}{2}(BC)$
$DE=\frac{1}{2}(BP+PO+OQ+QC)$
$DE=\frac{1}{2}(2PO+2OQ)$
{ $\because$ P and Q are midpoints of OB and OC}
$DE=PO+OQ$
$DE=PQ$ …..(2)
Now $\triangle AOC$ , Q and E are the midpoints of OC and AC.
$\therefore EQ\parallel AO$ and $EQ=\frac{1}{2}AO$   …..(3) {Using mid-point theorem}
Similarly, in $\triangle ABO,PD\parallel AO$ and   $PD=\frac{1}{2}AO$   …..(4)
From equation 3 and 4
$EQ\parallel PD$ and   $EQ=PD$
From equation 1 and 3
$DE\parallel PQ$ and   $DE= PQ$
Hence DEQP is a parallelogram.
Therefore option (D) is correct.